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PFAFFIAN SCHUR PROCESSES AND LAST PASSAGE PERCOLATION IN A HALF-QUADRANT JINHO BAIK, GUILLAUME BARRAQUAND, IVAN CORWIN, AND TOUFIC SUIDAN Abstract. We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy-Widom distributed, GOE Tracy-Widom distributed, or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy- Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging correlation kernels of point processes where points collide in the scaling limit. Contents 1. Introduction 1 2. Fredholm Pfaffian formulas8 3. Pfaffian Schur process 13 4. Fredholm Pfaffian formulas for k-point distributions 23 5. Convergence to the GSE Tracy-Widom distribution 33 6. Asymptotic analysis in other regimes: GUE, GOE, crossovers 46 References 56 1. Introduction 2 In this paper, we study last passage percolation in a half-quadrant of Z . We extend all known results for the case of geometric weights (see discussion of previous results below) to the case of exponential weights. We study in a unified framework the distribution of passage times on and off diagonal, and for arbitrary boundary condition. We do so by realizing the joint distribution of last passage percolation times as a marginal of Pfaffian Schur processes. This allows us to use powerful methods of Pfaffian point processes to prove limit theorems. Along the way, we discuss an issue arising when a simple point process converges to a point process where every point has multiplicity two, which we expect to have independent interest. These results also have consequences about interacting particle systems { in particular the TASEP on positive integers and the facilitated TASEP { that we discuss in [2]. Definition 1.1 (Half-space exponential weight LPP). Let wn;m be a sequence of i.i.d. n>m>0 1 exponential random variables with rate 1 when n > m + 1 and with rate α when n = m. We 1 The exponential distribution with rate α 2 (0; +1), denoted E(α), is the probability distribution on R>0 such −αx that if X ∼ E(α), P(X > x) = e : 1 m w22 w11 w21 w31 n Figure 1. LPP on the half-quadrant. One admissible path from (1; 1) to (n; m) is shown in dark gray. H(n; m) is the maximum over such paths of the sum of the weights wij along the path. define the exponential last passage percolation (LPP) time on the half-quadrant, denoted H(n; m), by the recurrence for n > m, ( n o max H(n − 1; m);H(n; m − 1) if n > m + 1; H(n; m) = wn;m + H(n; m − 1) if n = m with the boundary condition H(n; 0) = 0. Remark 1.2. It is useful to notice that the model is equivalent to a model of last passage percola- tion on the full quadrant where the weights are symmetric with respect to the diagonal (wij = wji). Remark 1.3. If one imagines that the light gray area in Figure1 corresponds to the percolation cluster at some time, its border (shown in black in Figure1) evolves as the height function in the Totally Asymmetric Simple Exclusion process on the positive integers with open boundary condition, so that all our results could be equivalently phrased in terms of the latter model. 1.1. Previous results in (half-space) LPP. The history of fluctuation results in last passage percolation goes back to the solution to Ulam's problem about the asymptotic distribution of the size of the longest increasing subsequence in a uniformly random permutation [4]. A proof of the nature of fluctuations in exponential distribution LPP is provided in [29]. On a half-space, [5,6,7] studies the longest increasing subsequence in random involutions. This is equivalent to a half-space LPP problem since for an involution σ 2 Sn, the graph of i 7! σ(i) is symmetric with respect to the first diagonal. Actually, [5,6,7] treat both the longest increasing subsequences of a random involution and symmetrized LPP with geometric weights. That work contains the geometric weight half-space LPP analogue of Theorem 1.4. The methods used therein only worked when restricted to the one point distribution of passage times exactly along the diagonal. Further work on half-space LPP was undertaken in [38], where the results are stated there in terms of the discrete polynuclear growth (PNG) model rather than the equivalent half-space LPP with geometric weights. The framework of [38] allows to study the correlation kernel corresponding to multiple points in the space direction, with or without nucleations at the origin. In the large scale limit, the authors performed a non-rigorous critical point derivation of the limiting correlation kernel in certain regimes and found Airy-type process and various limiting crossover kernels near the origin, much in the same spirit as our results (see the discussion about these results in Section 1.3). 2 1.2. Previous methods. A key property in the solution of Ulam's problem in [4] was the relation to the RSK algorithm which reveals a beautiful algebraic structure that can be generalized using the formalism of Schur measures [33] and Schur processes [34]. RSK is just one of a variety of Markov dynamics on sequences of partitions which preserves the Schur process [17, 13]. Studying these dynamics gives rise to a number of interesting probabilistic models related to Schur processes. In the early works of [4,6] the convergence to the Tracy-Widom distributions was proved by Riemann- Hilbert problem asymptotic analysis. Subsequently Fredholm determinants became the preferred vehicle for asymptotic analysis. In a half-space, [5] applied RSK to symmetric matrices. Subsequently, [37] (see also [26]) showed that geometric weight half-space LPP is a Pfaffian point process (see Section 4.1) and computed the correlation kernel. Later, [21] defined Pfaffian Schur processes generalizing the probability measures defined in [37] and [38]. The Pfaffian Schur process is an analogue of the Schur process for symmetrized problems. 1.3. Our methods and new results. We consider Markov dynamics similar to those of [17] and show that they preserve Pfaffian Schur processes. This enables us to show how half-space LPP with geometric weights is a marginal of the Pfaffian Schur process. We then apply a general formula giving the correlation kernel of Pfaffian Schur processes [21] to study the model's multipoint distributions. We degenerate our model and formulas to the case of exponential weight half-space LPP (previous works [5,6,7, 38] only considered geometric weights). We perform an asymptotic analysis of the Fredholm Pfaffians characterizing the distribution of passage times in this model, which leads to Theorems 1.4 and 1.5. We also study the k-point distribution of passage times at a distance O(n2=3) away from the diagonal, and introduce a new two-parameter family of crossover distributions when the rate of the weights on the diagonal are simultaneously scaled close to their critical value. This generalizes the crossover distributions found in [25, 38]. The asymptotics of fluctuations away from the diagonal were already studied in [38] for the PNG model on a half-space (equivalently the geometric weight half-space LPP). The asymptotic analysis there was non-rigorous, at the level of studying the behavior of integrals around their critical points without controlling the decay of tails of integrals. The proof of the first part of Theorem 1.4 (in Section5) required a new idea which we believe is the most novel technical contribution of this paper: When α > 1=2, H(n; n) is the maximum of a simple Pfaffian point process which converges as n ! 1 to a point process where all points have multiplicity 2. The limit of the correlation kernel does not exist2 as a function (it does have a formal limit involving the derivative of the Dirac delta function). Nevertheless, a careful reordering and non-trivial cancellation of terms in the Fredholm Pfaffian expansions allows us to study the law of the maximum of this limiting point process, and ultimately find that it corresponds to the GSE Tracy-Widom distribution. We actually provide a general scheme for how this works for random matrix type kernels in which simple Pfaffian point processes limit to doubled ones (see Section 5.1). Theorem 5.3 of [38] derives a formula for certain crossover distributions in half-space geometric LPP. This crossover distribution, introduced in [25], depends on a parameter τ (which is denoted η in our Theorem 1.9) and it corresponds to a natural transition ensemble between GSE when τ ! 0 and GUE when τ ! +1. Although the collision of eigenvalues should occur in the GSE limit of this crossover, this point was left unaddressed in previous literature and the convergence of the crossover kernel to the GSE kernel was not proved in [25] (see our Proposition 6.12). The formulas for limiting kernels, given by [25, (4.41), (4.44), (4.47)] or [38, (5.37)-(5.40)], make sense so long as τ > 0.

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